лабораторная работа / мэмс / Модуль1

Askerova Leila Ahmedivna

1^st Module

of Electromechanical systems modeling

Card#1

1.Basic concepts of modeling

Each person in the daily life allocates from infinite variety of displays of world around only their small number and on this basis forms some substitutes of real objects allowing him to build his purposeful behavior. These substitutes are adopted for naming models.

Thus, we shall agree to understand that a model is a substitute of real object with those properties and relations, which are required for the decision of practical problems. Accordingly modeling we shall consider as a method of no direct cognition in which the object - model is in some incomplete conformity with the object - original.

It is said, that the model is adequate to the original if it truly reflects properties of the original interesting for us.

The second circumstance, which to a certain degree facilitates our existence in indefinitely various world, is based on its material unity shown in similarity (analogy) of the heterogeneous phenomena and processes. Analogy is understood like judgment of resemblance of several objects in the certain relations on the basis of similarity established for other relations.

Also it is interesting to note interrelation of modeling and information understood as data of influences, values of their parameters, changes of these parameters in space and in time, taken in a separation from a physical data carrier and from its power properties.

Mastering by the information, its transformation, storage and display are impossible without modeling, i.e. without display in some material environment (in a human brain, on memories of the COMPUTER, on the screen of the display or on a sheet of a paper).

At construction of models - analogues property of isomorphism (similarity) of the mathematical description of processes of a various physical nature is used.

As a rule, real technical systems unite various physical nature subsystems into there structure. At modeling such complex systems it is necessary to take into account interrelations of subsystems.

2. Calculation of transients in linear electric circuits by numerical method. Method of state variable.

Method of state variable

Let's consider a simple electric circuit

R L

e (t) C

At the moment t₀ the switch is being turned on and a transient is started. To estimate transient it is possible with the help of Kirchhoff equation in the instantaneous form:

, (1.1)

However, the solution of the equations containing integrals is a difficult problem. To exclude integrals from Kirchhoff equations act as follows. It is formed Kirchhoff equations, in which side by side with currents, voltages of capacitor elements are appeared as unknown.

Then the equation (1.1) is resulted as

(1.2)

It is necessary to add the equation connecting instantaneous values of current and voltage of capacity to the equation (1.2)

(1.3)

The equations (1.2) and (1.3) are transformed to a normal form united into system

(1.4)

If generally for a complex circuit composite from some inductive and capacitor elements, this system looks as:

, (1.5)

where x₁,x₂…x_n - state variables which are currents in inductive elements and voltage on capacitor elements.

Conclusion:

Application state variables allow to result instantaneous Kirchhoff equations to the system of first- order differential equations reduced to a normal form.

3.Calculate steady-state processes in given electric circuit by modeling in Matlab environment. Verify simulation results.

e(t)=20sin(ωt+π/3)

R₁=2Ω;

R₂=1Ω;

R₃=5Ω;

L=0.002H;

f=50Hz.

Mathematical model:

Figure 1.2 – Impedances of the scheme

Directed graph of the scheme:

Figure 1.3- Directed graph of the scheme

So the branch here is just one. And there are two chords. It means that 1 equation should be composed according to the 1^st K`s law and two more are to be composed according to the 2^nd one.

2. Table of equations.

Table 1.1 – Table of coefficients of equations

	I1	I2	I3
I	Z1	0	Z3	E
II	0	Z2	-Z3	0
b	1	-1	-1	0

2. Block-scheme of the designed program

Module

f, R1,R2,R3, L, ph

Input data

Electro motive force

Impedance of branches

|M|

|F|

|I|=M\F

Modules and angels

Display

Currents,

Step, initial data

current

11

Plot the graph of currents

K=2:201

Program that executes the task

Check the errors

END

Errors: eps, eps1,eps2,eps3

END

t(k)=t(k-1)+h

current

%Initial data

f=50;R1=2;

R2=1;

R3=5;

L=2e-3;

ph=60;

w=2*pi*f;

XL=w*L;

E=100/sqrt(2)*exp(ph*pi/180*i);

Z1=R1;Z2=R2+i*XL;Z3=R3;

%Matrix of equations

M=[1 -1 -1;

Z1 0 Z3;

0 Z2 -Z3 ];

F=[0;E;0];

I=M\F;

disp(['I1=',num2str(I(1,1))]);

disp(['I2=',num2str(I(2,1))]);

disp(['I3=',num2str(I(3,1))]);

I1M=abs(I(1,1))

I2M=abs(I(2,1))

I3M=abs(I(3,1))

phI1=(angle(I(1,1))/pi*180)

phI2=(angle(I(2,1))/pi*180)

phI3=(angle(I(3,1))/pi*180)

h=1/(10^2*f);

t(1)=0;

i1(1)=current(I1M,f,phI1,t(1));

i2(1)=current(I2M,f,phI2,t(1));

i3(1)=current(I3M,f,phI3,t(1));

for k=2:201

t(k)=t(k-1)+h;

i1(k)=current(I1M,f,phI1,t(k));

i2(k)=current(I2M,f,phI2,t(k));

i3(k)=current(I3M,f,phI3,t(k));

end

%Output data

plot(t,i1,'x',t,i2,'+',t,i3);

title ('Currents in the branches of the scheme')

xlabel('time');

ylabel('Current');

legend('I1','I2','I3',-1);

grid on;

%Errors

eps21=E-Z1*I(1,1)-Z2*I(2,1);

disp(['error main contour eps2=', num2str(eps21)]);

Listing of the program

>> module

I1=15.132+19.0049i

I2=14.1137+14.3595i

I3=1.0183+4.6455i

I1M =

24.2933

I2M =

20.1343

I3M =

4.7558

phI1 =

51.4727

phI2 =

45.4945

phI3 =

77.6364

error 1st contour eps2=-1.7764e-015

Figure 1.4 – results of simulation